Simplify the following expression and state the condition under which the simplification is valid. $x = \dfrac{a^3 - 2a^2 - 15a}{-5a^2 - 40a - 75}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ x = \dfrac {a(a^2 - 2a - 15)} {-5(a^2 + 8a + 15)} $ $ x = -\dfrac{a}{5} \cdot \dfrac{a^2 - 2a - 15}{a^2 + 8a + 15} $ Next factor the numerator and denominator. $ x = - \dfrac{a}{5} \cdot \dfrac{(a + 3)(a - 5)}{(a + 3)(a + 5)}$ Assuming $a \neq -3$ , we can cancel the $a + 3$ $ x = - \dfrac{a}{5} \cdot \dfrac{a - 5}{a + 5}$ Therefore: $ x = \dfrac{ -a(a - 5)}{ 5(a + 5)}$, $a \neq -3$